MONOGRAFIES 6
ENRAHONAR
M
Antoni Malet
From Indivisibles to Infinitesimals Studies on SeventeenthCentury Mat...
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MONOGRAFIES 6
ENRAHONAR
M
Antoni Malet
From Indivisibles to Infinitesimals Studies on SeventeenthCentury Mathematizations of Infinitely Small Quantities
Universitat Autonoma de Barcelona Servei de Publicacions Bellaterra, 1996
DADES CATALOGRAFIQUES RECOMANADES PEL SERVEI DE BIBLIOTEQUES DE LA UNIVERSITAT AUTÓNOMA DE BARCELONA
Malet, Antoni ('ram Indivi,ibles to Infinitesimals : Studies on SeventeenthCentury Mathematizations of Infinitely Small Quantities. (Enrahonar. Monografies : 6)
ISBN 8449005205
1. Universitat Aut6noma de Barcelona. Departament de Filosofia (BeIlaterra) [1. Universitat Aut6noma de Barcelona. Servei de Publicacions (Bellaterra) lll. Col·lecció l. Infinit (Matematica) Historia S. XVII 2. Caleul Historia S. XVII 511"16"
Coberta Loni Geest & Tone H,lVerstad
Composició i IiImació MCMXCII. S.A. Passatge de L1ívia. 39 entresol 08041 Barcelona
Edició i impressiú Universitat Autonoma de Barcelona Scrvci dc Publlcacions 08193 Bellaterra (Barcelona l. Spain TeL (93) 581 1022. Fax (93)581 20m
ISBN 8449005205 Diposit legal: B. 42411996 Impres a Espanya. Printed in Spain
Imprcs en paper ecologic
A la Clara
CONTENTS 9
PREFACE
CHAPTER l. THE SEVENTEENTHCENTURY METHOD OF INDIVISIBLES
11
Bonavenrura Cavalieri's Indivisibles. Indivisibles and «the analyrical an of quadrature». Limirs in Geomeuical Oprics.
CHAPTER 2. REMAKING INDIVISIBLES: PASCAL, BARRO\X!, WALLIS
23
Torricelli. Pascal. Barrow. Wallis. Discussing the Hom Angle. Infiniresimals in Mathemarical Proof
CHAPTER 3. OPPOSING INDIVISIBLES: HUYGENS, GREGORIE, NE\X!TON
51
Boulliau. Huygens. James Gregorie on Quadratures. Transformarjon Theorems. Newton. Newton and Gregorie on Cavalieri's PrincipIe.
CHAPTER 4. JAMES GREGORIE'S «SOME GENERAL PROPOSITIONS OF GEOMETRY»
75
Inrroduceion. Description of rhe Manuscrit. Editorial Convenrions. '1' ranslarion. CHAPTER 5. METHODS OF TANGENTS ca.
1670
101
Inrroduction. FcrmatBeaugrand's Method ofTangents. The Kinematic Method ofTangents. Gregorie's Second Method ofTangents. Analyric Computation ofTangenrs. Barrow's Method «withour Ca!cuJarion... Tangents to rhe Cycloids. A Formula «'1'0 Change the Variable... Tangents to Curves ofTrigonomeuic Lines. Newton's Merhods ofTangents around 1670.
CHAPTER 6. INFINITES ANO INFINITESIMALS IN SEVENTEENTHCENTURY NATURAL PHlLOSOPHY
137
The Atomisation of Physical Effects. Bodies, Fluids and Mechanical Explanations. In and Out ofTheological Debates.
REFERENCES
157
PREFACE
Ir is only fair to say that the more powerful and sophisticated techniques of quadrature of the late seventeenth century wt!re largely indebted ro Cavalieri, particularly ro the program he set forth in his 1635 Geometria indivisibilibus continuorum nova quadam ratione promota, and to the early practitioners of the method of indivisibles, particularly Torricelli. They introduced general rreatments for curves and surfaces. They emphasized the use of general theorems ruling the transformation of geomerric figures as essential rools for determining their dimensions. They developed the insight that the equivalence of figures could be proved by comparing their indivisible elements, and suggested that a geomerric quantity could be determined as the sum of an infinite number of infinitely small parts. That all this is essential background to understanding the great mathematical contributions of the seventeenth century is well known. There is, however, substantial disagreement as ro the demonsrrative status the method of indivisibles held during the century. Furthermore, while indivisibles and infinitesimals came ro be interchangeable words for most mathematical practitioners, particularly after 1650, originally two very different notions were behind them. Cavalieri's genuine indivisibles (points, lines, an so on) were soon discarded, ro be substituted by infinitesimals. This essay analyzes some seventeenth-century texts closely related either to the changing notion of indivisibles or ro the status of the method itself. Ir is suggested here that the how and why of the rransformation of indivisibles into infinitesimals are related ro the hightened status acquired by the method of indivisibles among legitimate mathematical techniques. Chapter one, largely introducrory in character, reviews the first attacks and main objections raised against the method of indivisibles. Chapter two studies the main answers ro them, focusing on texts by Isaac Barrow and John Wallis. Chapter three compares the views of Christiaan Huygens, James Gregorie and Isaac Newron on the soundness of the method of indivisibles, and reviews some of the attempts to circumvent the flaws they discovered in it-including Ismael Boulliau's. Chapter four contains an annotated English version of James Gregorie's «Some General Propositions of Geometry»-an unpublished manus-
Indivisibles
(O
Infinitesimals
Anroni Maler
cript setting forth an Archimedean construction of the basic results of the method of indivisibles. This piece has the additional interest of con taining results closely similar to the mathematicallemmas opening Newton's Principia mathematica. Chapter five is devoted to the methods of tangents produced just before the publication of the first essays on the Leibnizian calculus, paying special attention to the role of infinitesimal notions in them. Finally, Chapter six reviews the role of the method of indivisibles in its philosophical con texto It is argued in conclusion that the main strength of the method was its dovetailing with key elements of the mechanical philosophy. In the chapter of acknowledgements I must gratefully mention the generous support fram the Diputació de Barcelona that has made this publication possible. Josep Montserrat, whose general support and encouragement in matters academical is a pleasure to acknowledge, eagerly received my manuscript and published it within the collection «Monografies» of Enrahonar. Material in Chapters 2 and 5 has previously appeared in «Barrow, Wallis, and the Remaking of SeventeenthCentury Indivisibles» (Centaurus, 38, 1995), and in «James Gregorie on Tangents and the 'Taylor' Rule for Series Expansions» (Archive jOr History 01Exact Sciences, 46, 1993, 97137). I wish to thank the publishers for kind permission to partially reproduce here these texts. Kind permission fram Edinburgh University Library to publish an English translation of James Gregorie's Latin manuscript is gratefully acknowledged. Research has been carried on in the University of Edinburgh Library, the Bodleian Library, and during a twomonth visit to the Munich Deutsches Museum in the summer of 1991 in the State Library of Bavaria. The staffs of these institutions are sincerely thanked for their help and kindness. Financial support from the Spanish Ministerio de Educación (Research Project PB900692) is also gratefully acknowledged. For comments and discussions on different parts of this book I am indebted to e.e. Gillispie, M.S. Mahoney, and K. Andersen. My thanks to Guillem and Jordi for their noisy playfulness and to Clara for her companionship.
Sto Cugat del Valles, September 1995
CHAPTER 1
The Sevenreenrh-Cenrury Merhod of Indivisibles
After the thirteenth and fourteenth centuries, when the notion that continua are composed of indivisibles was a minority opinion, the sixteenth century saw the revival of indivisibilism and the irruption of the troublesome notion of actually infinite space. The novel understanding of the continuum linked to these changes has not been yet properly explored, yet we know that attacks on the Aristotelian continuum carne from more than one direction. Innovative ideas carne from sixteenthcentury Aristotelians such as Ruggiero (who was lecturing on Aristotle's Physics in the Collegio Romano in the early 1590's). He considered both partes divisibiles and partes indivisibiles of the continuum, «thus conferring on indivisibles the status of parts also.» 1 Authors taking part in, or influenced by the NeoPlatonic reviva!, like Patrizi's or Bruno's, also abandoned the Aristotelian continuum. Renewed interest in Democritean and Stoic ideas played a part too. While these authors by no means agreed on their arguments nor on their doctrines, they variously defended the existence of a minimum of space, the existence of indivisibles of different sorts, and that the continuum was composed by its indivisibles. 2 How well mathematical practitioners accepted these ideas, we do not know. We have strong indications that mathematical magnitudes that were incomparable or incommensurable (because they were too big or too small) with finite magnitudes did come to play an ever growing role in sixteenthcentury mathematics and natural philosophy. Whether mathematicians agreed or not on the soundness of infinitely large magnitudes, Copernicus could use a clo-
1. W.A. Wallace, «Traditional Natural I'hilosophy», in C.B. Schmitt et al eds., Cambridge History ofRenaissance Philosophy (Cambridge: Cambridge University I'ress, 1988),201235, p. 214216 (quotation comes from p. 216). 2. ].E. Murdoch, «lnfinity and continuity», in N. Kretzmann, A. Kenny,]. I'inborg eds., Cambridge History ofLater Medieval Philosophy (Cambridge: Cambridge University I'ress, 1982),564591, p. 567584; E. Cassirer, Elproblema del conocimiento en la filosofia yen la ciencia modernas, 4 vol., W. Roces transo (Mexico D.F.: FCE, 1986; 1st German ed., 1906), l, 276288;]. Henry, «Francesco I'atrizi da Cherso's Concept of Space and its Later lnfluence», Annals ofScience, 36 (1979), 549573.
12 From Indivisibles ro Infinitesimals
Anroni Malet
sely related notion to make a counterargument against the objection that no parallax can be observed. While he assened the finite dimension of the radius of the fixed stars orbs, he took the radius of the eanh orbs to be incommensurable with ir. When taken twice, or several times (aliquotiens) , indivisibles (minimis corpusculis ac insectilibus, quae atomi vocantur) or atoms do not add up to visible objects, and yet they can be multiplied to such an extent that at last they make up a visible magnitude (apparentem magnitudinem): The same [can be said] of rhe place of the earth. Alrhough ir is nor the cenrer of rhe world, yer rhe disrance ro ir is srill incomparable, particularly wirh rhe sphere of the fixed stars."'
The relationship between finite magnitudes and their points or indivisibles raised questions substantial enough to be used in the skeptical debate of the second half of the sixteenth century. In a letter to Clavius of a bout 1574 the skeptic, Francisco Sanches, was to highlight the doubts this issue casts on the soundness of mathematical foundations. Mathematics cannot be a model of scientia, Sanches argued, not only because it is not causal, but because even its very principIes are doubtful. «For example, one supposes that there are points, but their existence and manner of being may be doubted. And the same applies to lines and surfaces.»4 We do not know how sixteenthcentury intellectual developments and contemporary changes in the social status of mathematics and mathematicians may have predisposed sorne firstrank mathematicians to embrace a non Aristotelian continuum. By the turo of the seventeenth century, however, Johannes Kepler had very successfully used infinitesimal notions in problems involving quadratures and cubatures, and others soon followed. During the first two or three decades of the century a sizable group of mathematicians developed mathematical techniques of quadrature that ro a lesser or greater extent rested on the notions of indivisibles and infinitesimals. 5 Bonaventura Cavalieri's Indivisibles The origins of infinitesimals, one of the most fruitful as well as debated mathematical notions from around 1600 to around 1800, can fairly accurately be
3. «Ita quoque de loco rerrae, quamvis in cenrro mundi non fuerit, distantiam ramen ipsam incomparabilem adhuc esse praesertim ad non errantium srellarum sphaeram.» See N. Copernicus, De revolutionibus orbiunJ caelestiunJ libri sex (F. Zeller, K. Zeller eds., München: R. Oldenburg Verlag, 1949), p. 17. 4. F. Sanches, That Nothing is Imown (E. Limbrich ed., D.F.S. Thomson trad., Cambridge: Cambridge University Press, 1988), p. 4950 (1 am quoring Limbrick's paraphrase of Sanches's words). 5. M. Baron, The Origins ofthe Infinitesimal Ca!culus (New York:D?ver, 1969), p. 108ff; A. Koyré, «Bonaventura Cavalieri er la géomérrie des conrinusn. in Etudes d'histoire de la pensée scientifique (Paris: Gallimard, 1973),320349, p. 322.
Chapter 1
The SeventeenthCentury Method of Indivisibles 13
traced back to Bonaventura Cavalieri's 1635 Geometría índivisibilibus continuorum nova quadam ratione promota, the book that first introduced indivisibles into mainstream seventeenthcentury mathematics. This is by no means to say that indivisibles or infinitesimals were not used before, but that Cavalieri first offered a «rigorous» geometrical construction of the powerful intuition that sees geometric magnitudes as made up of infinitely many small constituents. Up to the seventeenth century infinites and infinitesimals had no proper place in mathematical discourse, and Stevin and Kepler, whose contributions appeared in works that properly speaking belong not to pure mathematics, made no attempt to bring their new notions and methods within the domain of Euclidean canons. This is what makes Cavalieri's work to occupy a special place in the introduction of infinitesimal methods. 6 In possession of an ecclesiastical education in philosophy and theology, Cavalieri was well aware of the centuries old debate on the nature of the continuuma debate that provided many arguments against the continuum being made up by, or equal to, the reunion of its «minimal pans» or indivisible elements. He allowed himself only the son of indivisibles that classical geometry and philosophical tradition knew of: points, lines, and surfaces. These indivisibles Archimedes did use to good effect in his «mechanica]" method of quadratures and cubatures.7 Within the mathematical principIes Euclid and Archimedes set fonh, indivisibles cannot be «pans» of the mathematical object they belong tobecause comparability and the quality ofbeing a pan go together. If indivisibles were pans, they would be comparable and homogeneous with their whole. An «infinitely small paw> was therefore a contradiction in terms, words designating objects that cannot existin the same sense that triangles with one side longer than the other two together could not exisr. Apparently embracing these views, Cavalieri carefully refrained from stating
6. G. Cellini, «Gli indivisibili nel pensiero matematico e filosofico di Bonaventura Cavalieri)" Periodico di matematiche, 4 ser., 44, 1966, 121; «Le dimostrazioni di Cavalieri del suo principio», Periodico di matematiche, 4 ser., 44, 1966, 85105; L LombardoRadice, Inrroduction ro Geometria degli Indivisibili di Bonaventura Cal'alieri, L LombardoRadice ed. (Torino: Eunice, 1966); K. Andersen, «Cavalieri's Method of Indivisibles», Archive fór History ofE>:act Sciences, 31, 1985,291367; E. Giusti, Bonaventura Cavalieri and the Theory oflndivisible,- (Bologna: Cremonese, 1980); «Dopo Cavalieri. La disCllssione sugli indivisibili», in Atti del Convegno «La Storia de!!e Matematiche in Italia».Cagliari. 29-30 settembre e 1 ottobre 1982 (Cagliari: Universita di Cagliari, n.d.), 85114; E. Festa, «Quelques aspects de la conrroverse sur les indivisibles», in M. Bucciantini, M. Torrini eds., Geometria e atomismo ne!!a scuola galilea na (Firenze: Leo S. Olschki, 1992), 193207; D.M. ]esseph, «Philosophical Theory and Mathematical Practice in the Seventeenth Century», Stud. Hist. Phi! Sci., 20, 1989,215244; F. de Gandt, «Cavalieri's Indivisibles and Euclid's Canons», in P. Barker, R. Ariew eds., Revolution and Contilluiry. Essays in the History and Philosophy ofModern Science (Washington: The Catholic Universíty of America Press, 1991), 157182. 7. E.]. Dijksterhuis, Archimedes (Princeton: Princeton Universiry Press, 1987, C. Dikshoorn trans., with a bibliographic essay by W.R. Knorr; Isr ed. 1938), 313336. Of course Cavalieri did not know of Archimedes's «mechanical theorems», which were published only in 1912.
14
From Indivisibles
QI S,
T
Antani Malet
ro Infmitesimals
V R
v
]
M
A
;1/N
IK ;h
p
L
B
G
l
lE
D
e
H
Figure 1.
that magnitudes were made up by the reunion or addition of their indivisibles. To circumvent this difficulty, however, he introduced mathematical objects also foreign to the Greek classical tradition, those of «all the lines» and «aH the planes». Anachronistically speaking, «aH the lines» of a a given figure is the set of aH the straight line segmenrs conrained within a figure and paraHel to a given line (the reguia). These notions provided Cavalieri with means for proving the socalled Cavaiieri's principie along with several quadratures. According to this principie (here exemplified only in the case of surfaces, but of application to any twO geometrical magnitudes whose indivisibles can be compared), if two figures ABT, ABC (see Figure 1) are such that for any straight line NKO paraHel to the base TC the ratio NK : KO is the same, then the twO surfaces ABT, ABC also are in the same ratio. Cavalieri offered three proofs of this result, but the three of them were deemed incomplete by his contemporaries. Cavalieri's flrst proof is given in Proposition 4, Bk. 2, of his Geometria indivisibilibus. Cavalieri was careful enough ro address the question of whether a ratio between «all the lines» of a figure and those of another could possibly existo The affirmative answer ro this question is proved in the opening Proposition 1 of Book 2. Then he proved in Proposition 3 that «aH the lines» are in the same ratio as the figures. The proof, carried on by «superpositioll», may entail an infinite processas his critic Guldin poinred out. From Propositions 1 and 3 his principie easily foHows. Cavalieri's second and third praofs, both avoiding the use of the troublesome notion of «aH the lines», are given together in Proposition 1 of Book 7 of the Geometria indivisibilibus. The second proof depends on his attempt to demonstrate, also by «superposition», the equality of two figures whose «lines» are one by one equal. His contemporaries found the proof objectionable as well. The third proof was added «Since the preceding proposition is a most momentous one and a differem oo')
The SeventeenthCentury Methad afIndivisibles 15
Chapter 1
manner, not differenr from Archimedes's style, to proof its first pan has come ro my mind».8 As the words suggest, this proof uses an exhaustion technique. Modern commentators, including CeHini and G. Castelnuovo, have defended its soundness, even though Cavalieri's contemporaries were not convinced by it either. 9 As is weH known, Cavalieri's indivisibles were soon substituted by infinitesimals, and the method of indivisibles grounded on the assumption that the surfaces of figures, the volumes of solids, and the lengths of lines are equal to aH the indivisibles they conrain taken rogether. 10 Provided that the operation of «taking together» is understood as sorne son of addition, Cavalieri's principie can be derived from the assumption just menrioned. Let us assume a
m
b
n
for all the ordinates a = NK, b = KO in Figure l. lf for a finite number of /s it is a. : b. = m : n, then we obviously have Ia. : Ib. = m: n. We may then concluJe that i I i 1 AH the as
m
AlI the b's
n
by analogy with the addition of a finite number of magnitudes. Two difficulties arise. One, which did not seem to bother seventeenthcentury mathematicians, is the very notion of a sum that involves an infinite number of terms, and the demonstration of a rule involving such a sumo The second and historicaHy more meaningful difficulty in this approach is that the classical notion of indivisible must be set aside. lf «aH the lines» is to yield the surface of a figure, and if «aH the lines» results fram sorne son of addition of the lines, the lines must be understood to be rectangles whose heights are ordinasee, in the te lines and whose bases are infmitesimal segments. As we セィ。h middle decades of the century indivisibles were explicitly transformed into infinitesimals by sorne leading mathematicians, including Pascal (around 1658) and Barrow (around 1665, first published in 1683).
8. B. Cavalieri, Geometria indivisibilibus continuorum nova quadam mtione promota (Bologna, 1653), p. 488. 9. For very good detailed accoums ofCavalieri's proofs, see Andersen's «Cavalieri's Method ofIndivisibles», and Ce]]jnj's «Gli indivisibili ne! pensiero matematico e filosofico di Bonaventura Cavalieri», and «Le dimostrazioni di Cavalieri de! SUD principio»; see also LombardoRadice's editorial notes ro his Italian edition of Cavalieri's book. 10. Cavalieri never asserted such a thing, bur T orricelli and many others did; see Andersen, «Cavalieri's Method of Indivisibles», passim.
16
From Indivisibles ro Infinitesimal,
Antoni Maler
Indivisibles and «the analytical art of quadrature» From the very beginning, fruitfulness was recognized as the major virtue of the method of indivisibles. However, while it was recognized to be a powerfui analytical tool, its demonstrative status was much debated in the decades following the publication of Cavalieri's Geometria. In this period Cavalieri and Torricelli worked hard to legitimate the use of indivisibles. Generally speaking, they managed to ensure a place for them in the mathematicians's armory, but only as a heuristic weapon. While proudly stressing that the method of indivisibles was very fruitful indeed, they assumed a rather defensive stance towards the soundness of its foundations. Cavalieri and Torricelli compared the method of indivisibles to Algebra and to the analytic art that the ancients must have used, and left hidden, to make their discoveries. This was a powerful theme in the early seventeenth 11 century, as M.S. Mahoney convincingly argued in his srudy on Fermat. Cavalieri, for instance, compared the aggregates or congeries of lines to irrational arithmetical quantities, and used the algebraist's handling of irrational quantities as justification for his notion of «all the lines»: 1 have used an artifice [artijicio] similar ro rhar often used by Algebraists for solving problems. For, however ineffable [ineffabilisJ, absurd and unknown the roots of numbers be, they none the less add, subtract, multiply and divide them; and they are convinced of having sufficiently met their obligation when they have succeeded in finding out from the given problem the result which was required. Not differently, therefore, it is legitimate for me ro make usein arder ro investigate the measure of continuaof the congeries of indivisibles, either of lines ar planes, ... for if they are nameless [innorninabilisJ, absurd and unknown with respect to their number [ofindivisiblesJ, rheir magnirude [rnagnitudo] is enclosed in welldefined limits. 12 The comparison with the analytical art of the ancients is explicit in Torricelli: But this Geometry of rhe indivisibles is hardly a new invention .,. [.] 1would rather believe rhat i:he aneient Geometers did use this merhod in arder ro discover their most difficult Theorems, although they preferred taking a different path ro prove themeither to hide their secret art, ar else not ro allow invidious detractors any occasion for criricism.]3
11. M.S. Mahoney, The Mathematical Career ofPierre de Fermat 1601-1665 (Princeron: Princeron Universiry Press, 1973), p. 2648. 12. Geometria indivisibilibus continuorum nova quadam ratione promota (Bologna, 1653), b2 recto (all references are made ro rhis edirion). Translarions are my own, bur 1 have benefired fram L. LombardoRadice's lralian rranslarion, Geometria degli [ndivisibili di Bonaventura Cavalieri (Torino, 1966). 13. E. Torricelli, Opere, 3 vol., G. Loria eta1ed. (Faenza, 1919),11,139. Translarions are my own.
Chaprer I
The ScventeenthCentury Method of Indivisibles 17
Among the many advantages of the «geometry of the indivisibles», Torricelli listed its eHiciency in providing short proofs and its heuristic power: Be ir as it may, rhe rruth is that this geometry provides an exuaordinary short cut ro rhe faculty of invention and thar it supporrs innumerable very difficult rheorems by short, direct, posirive [aff¡'rrnativus] demonstrarions, which can hardly be achieved by means of the doctrine of the ancients. In fact, rhis geomerry is rhe ttuly royal road for sharp mathematicians, and Cavalieri, aurhor of wonderful invenrions, was rhe [¡rst to discover it and ro explain it for [he public benefit.]4 William Oughtred, the influential English mathematics teacher, and someone not involved in the creation of the method of indivisibles, did not faíl to notice and stress its power. He says, in his 1645 answer to one Keylway interested in the quadrarure of the circle, 1 ... am induced to a better confidence of your performance, by reason of a geometricanalytical art ar pracrice found out by one Cavalieri, an Italian, of which about three years since 1 received information by a lerter from Paris, wherein was praelibated only a small taste thereoE, yer so that 1 divine grear enlargement of the bounds of the marhematical empire will ensue.]5 During the middle decades of the seventeenth century recognition that indivisibles involved conceptual problems was more often than not accompanied by the notion that they were very much needed to deal with otherwise intractable problems. Cavalieri himself, however hard he tried to provide sound foundations for his method, could not avoid recognizing that the new principies he was using would not convince every mathematician. He was thus led to advance the notion that concerns about foundations are philosophical rather than mathematical. He also used the argument that the basic notions and techniques of the method of indivisibles must be sound because the results they provide agree with those found by other means. He says, in his Geometria indivisibifibus (the argument reappears twelve years later in his Ey;ercitationes geometricae sex): 1 do nar think ir is useless ro no rice .. , that mosr of rhe things that were disclosed by Euclid, Archimedes, and others, have also been demonstrared by me, and rhar my conclusions agree perfectly with their conclusions. This must be an evident indication that 1 have gathered rrue things in rhe basic principies [o!rny rnethodJeven though 1 know that from false principIes rruths can be sophisrical1y deducedo But that that had happened ro me in so many conclusions [otherwúej proved geometrically, 1would deem ro be absurd. 16 14. [bid., p. 140. 15. Oughrred ro Roben Keylway, [Ocrobre 1645], in S.]. Rigaud ed., Correspondence ofscientiflc men ofthe seventeenth century (2 vols., Oxford, 1841, repr. Hildesheim: Georg Olms, 1965),1,65. 16. Geometria indivisibilibus, p. 112; ExereitiUiones geometricae sex (Bologna, 1647), p. 195.
18
From Indivisibles
to Infinitesimals
Antoni Malct
Let us stress, incidental!y, that Leibniz was to use a similar argument in the early 1690s to foster the cause of his «nouvel!e analyse des infinis». As a positive test of the sour.dness of his infinitesimal analysis, he produced the perfect agreement between results found through it and those found by other means, «quoiqu'il n'y ait eu aucune communication entre les auteurs des solutions; ce qui est une marque de la vérité».17 We now know that Cavalieri's arguments failed to convince most of his contemporaries. Overlooking or rejecting his subtile treatment of the col!ection of indivisibles induded in any magnitude as a mathematical object distinct from the magninide itself, most mathematicians assumed Cavalieri's method of indivisibles ro entail an aromistic view of geometrical magnitudes. Wel!known, centuriesold arguments against mathematical atomism did also play a role in the development of the method through the seventeenth century. As is well known, the most direct and formal attacks on the method of indivisibles carne from two Jesuits, Paul Guldin (15771643) and André 1'acquet (1612 1660). 1'he mathematical part of Guldin's attack (he also blamed Cavalieri for plagiarizing Kepler and Soverus) rests on his rejeetion of «al! the lines» and «al! the planes» as legitimate mathematical objects susceptible to have a ratio to one another. 1'he main argument in 1'acquet's 1651 criticism was that no magnitude is an aggregation of its heterogeneous indivisibles. 18 Al! competent mathematicians, however, knew that no kind of geometrical continuum could be conceived as composed of indivisibles heterogeneous to the continuum itself Arguments against, for instance, a line being but an aggregate of its points can be traced back to Arisrotle, and they had lost none of their strength and cogency in the seventeenth century. Let us stress, therefore, that mathematicians opposing the method of indivisibles were not the only ones aware of the logical dangers lurking in an uncritical acceptance of indivisibles. Highly articulate expositions of the need of banning indivisibles from geometry are found, however confusing that may appear, in at least three very distinguished practitioners of «indivisible» techniques, Isaac Barrow, Blaise Pascal and John Wal!is. 1'0 account for this apparent contradiction we must pay attention to the changing meaning of the word
17. G.W. Leibniz, «De la chainette, ou solurion d'un probleme fameux, proposé par Galilei, pour servir d'essai d'une nouvelle analyse des infinis, ... " Uournd des S,avam, 1692), in Mathematische Schriften, el. Gerhardr ed. (Hildesheim, 1962), V, 258263, p. 260. 18. Guldin's argumenrs are reproduced verbarim in exercitatio III of Cavalieri's Exercitationes geometrú;ze sex, p. 177241. T acquer's much more brief ones are found in his Cvlindricorum et ammlarium libri IV (Anrwerp, 1651), p. 2124. See E. Giusri, Borzaventura Cavalieri and the Theory o/Indivisibles, p. 5565 and 7376; and «Dopo Cavalieri. La discussione sugli indivisibili", in Atti del Convegno "La Storia del/e Matematiche in Italia". Cagli,zri, 2930 settembre e 1 ottobre 1982 (Cagliari:Universira di Cagliari. n.d.), 85114; E. Fesra, "Quelques aspecrs de la conrroverse sur les indivisibles", in M. Buccianrini, M. Torrini eds., Geometria e atomismo nel/,z scuola galileana (Firenze: Leo S. Olschki, J 992), 193207. See also H. Bosmans, «André T acquen>, in Académie Royale de Belgique, Biographie nationale (Bruxelles, 19261929), XXIV, mis. 440464.
Charter 1
The SeventeenthCentu'ry Method of Indivisibles 19
,ándivisible,). Mathematicians of the middle decades of the seventeenth century explicitiy abandoned the old notion of heterogeneous indivisibles and gave the leading role to infinitesimals, a quite new notion occupying an intermediate position between indivisibles a Lancienne and finite geometrical magnitudes. As we shal! see, they did so partiy by giving a new meaning ro one of the main Arisrotelian arguments against dassical atomismthe infinite divisibility of the geometrical continuum. Interpreted as providing an actuaL infinite division, this argument supported the view that a finite line can be divided into an infinite number of infinitely sma11 pam. Interestingly enough, the substitution was not registered at the rherorical level, tor the expression «method of indivisibles» was ro be used through the century. 19 In common with the old indivisibles, infinitesimals had the property that no finite number of them taken rogether could be equal to or greater than any finite magnitudein modero jargon, infinitesimals were nonArchimedean magnitudes. But Cavalierian indivisibles and Barrow's «indefinitely smal!linelets» (indefinite parvis LineoLis) differ in three essential aspects. Infinitesimals are divisibLe; that is, it makes mathematical sense to halve an infinitesimal. 1'hey are hornogeneous with the magnitude they originated from. In particular, by taking together infinitesimal surfaces, a larger surtace is gainedwhich wil! be infinitesimal itself, if their number be finite, or it wi11 be finite, if their number be infinite. Finally, infinitesimals can actually be reached by a process of infinite divisiona process which can never yield a point, say, starting from a linear segmento Ir has been said that the method of indivisibles encountered no real opposition during the seventeenth century. As this account has it, worries about rigor and sound reasoning were not central ro seventeenthcentury mathematical thought, and with the exception of1'acquet and Guldin no real mathematician positioned himself publidy against the new notions of Cavalieri and his fo11owers. Those who did, mostiy persons of no consequence, were motivated by conceros, philosophical or otherwise, of no importance to the mathematical community.20 Although this picture contains sorne truth, it is incomplete. For one thing, many firstrank mathematicians, induding Newton, Gregorie and Huygens, did worry about the foundations of the method of indivisibles. Newton and Gregorie did offer their particular version of a demonstration of Cavalieri's principie. On the other hand, the method of indivisibles was not straightforward in its application. Himself an advocate of indivisibles, Pascal was led ro use the method of exhaustion to setde a question in which two proofs, one by indi19. While rhe notíon of infiniresimals was clearlv introduced inro mainstream marhemarical rhoughr by rhe middle of rhe 17rh cenrury, rhe word «infiniresima[" is almosr never ro be found in sevenreenrhcenrury rexrs. More ofren rhan nor, rhe rerm used is "indivisible", The French also used "infinimenr perir". In Larin ir was common ro refer ro parrs rhar were ((infinice parvae» or (exiguae). 20. See E. Giusri, Bonaventura Cavalieri and the Theory o/Indivisibles (Bologna, 1980), p. 48. Compare wirh D.T. Whireside's "Parrerns of Marhematícal Thoughr in rhe ¡arer Sevenreenrh Cenrury", Archive ftr History o/Exact Sciences, 1(1960), 179388, p. 327.
20
From Indivisibles ro Infinitesimals
Antoni Malet
Chaptet I
visibles and the other by motion, yield conflicting conclusions. 21 As we shall see, the use of indivisibles could produce paradoxes involved enough to lead the young Huygens to consider the method of indivisibles completely unreliable. It must be stressed, however, that the status of the method of indivisibles improved markedly as the cenrury went on. We know that Fermat and Torricelli could appreciate the advantages of indivisibles as tools of discovery, but would use them only warily as tools of proof. In the second half of the seventeenth century we find many leading mathematicians, such as Pascal, Wallis, Sluse, and Barrow, willing to grant fulllegitimacy to the method of indivisibles. In the same period a growing number of authors asserted the essential eguivalence between proofs involving indivisibles and proofs patterned according to the classical exhaustion method. In Pascal's wellknown words, «tout ce gui est demontre par la veritable regle des indivisibles se demontrera aussi a la rigueur et a la maniere des anciens; ... l'une de ces methodes ne differe de l'autre gu'en la maniere de parien).22 This meant that the method of indivisibles had gained a new status: it was now deemed able to provide legitimate demonstrations, which was something that few mathematicians would have dared to say in the 1630s and 1640s. In the next chapter we shall find evidence that beneath this significant modification of the logical status of the method of indivisibles lies the substitution of infinitesimals for classical indivisibles.
The s・カ ョイ・ ョイィセc・ョエオイケ
Method of Indivisibles 21
L
o A
F
E
D T
G
e
Limits in Geometrical Opties
It has been said that seventeenthcentury mathematicians got entangled with infinitesimals because they failed to use the notion of mathematicallimit. Historically speaking, it is true that the vexed guestion of the métaphysique du calcul infinitésimal-to put it in eighteenthcentury lingo was definitely setded through Cauchy's definition oflimit in the first decades of the nineteenth century. The 610 definition of limit, however, has litde to do with the foundational guestions that worried seventeenthcentury mathematicians, with the notions they were handling, and with the mathematicallanguage they had at their disposal. Although in the seventeenth century mathematicallimits were of no conseguence in relation either to the solution of mathematical problems or to foundational matters, perhaps it is worthwhile recalling that seventeenthcentury geometrical optics did use a notion closely related to the modero notion oflimit. Let us assume that a family of rays LF, OE, ... (see Figure 2), the prolongation of which gather in sorne point D, is refracted by the surface AFE. Christiaan Huygens established that the point T, defined by taking TA : DA 21. «Lenre de A. Denonville a Monsieur A.D.D.S. en luy envoyanr la Demonstration a la maniere des Aneiens de I'Egalite des Lignes Spiraie et Parabolique» (10 Deeember 1658), in Oeuvres, VlII, p. 255ff. 22. B. Pascal, Oeuvres, P. Bourroux, L. Brunsehvieg, F. Gazier eds., (14 vols., Paris, 19041914), VIll, p. 352.
Figure 2.
egua! to the refraction index, is the actual or «real» gathering point after refraction. In order to demonstrate it he proves, first, that no ray reaches the axis AD between A and T. Secondly, that EA > FA implies CT > GT. And finally, that there are always rays that after refraction meet the axis AD at points whose distance to T is smaller than any given distance. 2.3 T is, therefore, the limit of all the intersections C, G, ... as the angle ADF gets smaller, and Huygens's characterization seems to be as good a (geometrical) definition of limit as it could be hoped for in the seventeenth century. Huygens, however, never carne to use a notion similar to it out of the context of geometrical optics. As we shall see, he was indeed worried about the cogency of indivisibles and infinitesimals. He used them to discover results, but was reluctant to use them in mathematical demonstrations and avoided them as much as he could. A similar consideration can be made in relation to Barrow's geometrical definition of optical images, although he was not reluctant to use infinitesimals. 24 For 23. C. Huygens, Dioptrica (1653), in Oeuvres completes de Christiaan Huygens (22 vol.. La Haye, 18881950), XIII:1, p. 18. 24 For Barrow's geometrieal ¡imits, see 1. Barrow, Lectiones XVIII ... in quibus opticorum phaenomenon genuinae rationes investigantur, ac exponuntur (1669), in The Mathematiml Works, W. WheweIl ed. (Cambridge, 1860), p. 5152.
22
Fmm Indivisibles ro Infinitesimals
Anroni Malet
limits to be of any use in solving the puzzle posed by infinitesimal notions, the objeets of the infirtitesimal ca!culus must be funetions expressed and defined analytieally, but those objeets were absent fram seventeentheentury mathematies.
CHAPTER2
Remaking Indivisibles: Pascal, Barrow, Wallis
As is well known Cavalieri's notions were urrerly misrepresented during the seeond half of the seventeentheentury, and Torrieelli one of Cavalieri's keenest followers has been partieularly blamed for this misrepresentation. 1 Examples showing how the word «indivisible» was misused in the seeond half of the seventeenth eentury eould easily be multiplied. K. Andersen, for instanee, pointed to Roberval as one among many who «prajeeted his own ideas into [Cavalieri's theoryJ; thus he intradueed the idea that it was based on infinitesimals». 2 A good example of the muddling of Cavalieri's ideas is pravided by John Wallis's following aecount. He explained in his 1685 Treatise o/Algebra that the «Geometry ofIndivisibles, or Method ofIndivisibles» was "First inttodueed by Bonaventura Cavallerius, in a Treatise by him Published in the Year 1635; and pursued by Torrieellius, in his Works Published in the Year 1644.» Wallis had no qualms about handling infinitesimal magnitudes, as if they involved no logieal diffieulty whatsoever, but he did stress that the soealled indivisibles should not be understood as points, lines, or planeswhieh, as we now know, is how Cavalieri undersrood them. Rather they had ro be substituted by infinitesimals: According ro this Method [oflndivisibles], a Line is considered, as consisting of an Innumerable Multitude ofPoints: A Surface, ofLines, ...: A Solid, ofPlains, or other Surfaces..... Now this is not ro be so undersrood, as if those Lines (which have no breadth) could 611 up a Surface; or those Plains or Surfaces, (which have no
l. K. Andersen, «Cavalieri's method of indivisibles», Archive ftr History o[Exact Sciences, 31 (1985), 291367; E. Giusti, Bonaventura Cavalieri and the Theory o[Indivisibles (Bologna: Cremonese, 1980); F. de Gandt, «Les indivisibles de Torricelli», in F. de Gandr ed., L oeuvre de Torricelli: science galiléenne et nouvelle géométrie (Nice: Les Belles Lettres, 1987), 147206; and «Cavalieri's Indivisibles and Eudid's Canons», in P. Barker, R. Ariew eds., Revolution and Continuity. Essays in the History and Philosophy o[Modern Science (Washingron: The Catholic Universiry of America Press, 1991), 157182. 2. K. Andersen, «The Method ofIndivisibles: Changing Understandings», in A. Heinekamp (ed.), 300]ahre «Nova methodus» von G. W Leibniz (1684 1984) (Wiesbaden: Steiner Verlag, 1986), 1425, p. 23.
,.. 24
From Indivisibles to Infinitesimals
Antoni MaJet
Chapter 2
B
thickness) could compleat a Solido But by such Lines are ro be undersrood, small Surfaces, (of such a length, but very narrow,) ... 3
And Wallis makes clear that in this context «very narrow» stands for infinitely narrow. Once we accept that important differences exist between Cavalieri's indivisibles and their putative descendants, the infinitesimals, it seems historically pertinent to try and answer sorne related questions: Why were infinitesimals introduced? How were they introduced? Why were they preferred ro classical indivisibles? The secondary literature, particularly the older one, mostly suggests that mathematical rigor and unassailable foundations were not of paramount importance during this periodthat the fruitfulness of the new methods and notions balanced, as it were, the logical deficiencies in their foundations. Wc would like ro suggest here that just the opposite may be the truth. That it was precisely because mathematicians cared a lot about questions uf rigor and foundations that they set out to modify Cavalieri's theory. By focusing on detailed discussions by Blaise Pascal, Isaac Barrow and John Wallis on foundational questioIlS re1ated to indivisibles and infinitesimals, we will demonstrate that at least for them, influential mathematicians all, the shift from indivisibles to infinitesimals was one ro which they devoted much thought. When they came eventually to embrace infinitesimals and reject indivisibles, they did so because they thought that the former offered much more sol id foundations than the latter. By substituting infinitesimals for indivisibles mathematicians provided a new status to the method of indivisibles, which thereby became a legitimate method of proof. 4 Torricelli, in answer to a numbet of paradoxes apparently entailed by Cavalieri's method, set out to modify Cavalierian indivisibles. 5 As F. de Candt has recently shown, Torricellian indivisibles could be compared by «quantity» and were reachable through infinite subdivision. 6 They are our starting puint.
3. J. Wallis, A Treatise ofAlgebra, bot/' Hirtorical and Practical (London, 1685),285286. 4. Compare with D.M. Jesseph, "PhiJosophical Theory and Mathemarical Praetice in the Seventeenth Century», Stud. Hist. Phi/. Sci., 20, 1989, 215244; and also with A. Malet, "Providing Foundations m New Methods of Quadraturen, in "Studies on James Gregorie (16381675)>> (Ph.D. Diss., Princemn University, 1989),216283. 5. Torricelli gathered an impressive collection of f¿1se mathematÍcal resu1rs derived by using indivisibles. They are found in "Contro gl'infiniri», in E. Torricelli, Opere, 3 vol. (in 4), G. Loria, G. Vasurra eds. (Faenza, 1919), 12, p. 47ff, and in "De indivisibilibus», ibid., p.415ff. 6. F. de Gandt, «Les indivisibles de T orricellí», in F. de Gandt ed., L 'oeuvre de Torricelli: science galiléerme et nouvelle géométrie (Nice: Les BeHes Lettres, 1987), 147206; «L'evolutÍon de la théorie des indivisibles et l'apport de Torricelli», in Bucciantini, M. Torrini eds., Geometria e atomismo /leila scuola gaülea/la, 103118. See alsu E. Bortolotti, ,,1 progressi del metodo infinitesimale nell'opera geometrica di Evangelista Torricelli», Periodico di matematiche, 4 ser, 8,1928,1959.
...
Remaking Indivisibles: Pascal, Barrow, Wallis 25
pcセ
Q
A
R
D
s
e
Figure 1.
Torricelli To put Torricelli's thought in context let us review a paradox, ofren dealt with by both supporters and critics of indivisibles alike, that Cavalieri and Torricelli deemed particularly damaging. In the nonisosceles triangle ABC (see Figure 1) the two triangles BDA, BDC are different. Every line PQ parallel ro the base AC determines two equaflines PR, QS. As the line PQ moves parallel to the base and occupies every position from D to B, it determines an equal number of equal indivisibles PR and QS. Hence, «AlI the lines PR» '" «AlI rhe lines QS», and the triangles BDA and BDC ought to be equa!.? A second paradox, concerning concentric cirdes, is worth memioning because we shall see below how Barrow dealt with ir. Consider two concentric cireles and join the center to al! the poims of rhe larger circumference. It is obvious, says Torrice1li, that as many points will be determined by the intersccrion of all the radii with the smaller circumference as there are points in the larger one. Torriccl!i conduded from here that the points in the smaller circumference were «smaller» (sic) than those in the greaterindeed, the points were in the same proponiun as the diameters. 8 Torricelli, therefore, endowed indivisibles with different magnitudes and imroduced comparisons between them: That all the indivisibles be equal ro one another, that is, poims ro poims, lines (in width) ro lines, ... , is an opinion, in my view, not only hard to prove, but false.9
7. Cavalieri discusses this paradox in his 5 April1644leIter to Torricel1i, in Turricelli's Opere, IIJ, 1701; he returns to it in his 1647 Exercitationes geometriwe sexo CL 11. Bosmans, «Sur une concradiction reprochée a la théorie des "indivisibles» chez Cavalieri", Arma/e,· de la Société Scientifique de Bruxelles, 42 (192223),829. 8. T orricelli, Opere, 12, 320. 9. This, and rhe nexr fragments, must have been writren ca. 1645, see Opere, 12, 320.
26
Antoni Malet
From Indivisibles ro Infinitesimals
Chapter 2
Facing a paradox very similar ro one of the aboye, Torricelli explained away the different lengths of the sides AB and BC (in Figure 1) by introducing the idea that the points on each side had different lengths:
if this division is made, or if it is assumed ro have been made, an inhnite number of times, we would come ro have, instead of trapezoids, a line BC equal ro the line BA. 1 mean equal in quantity (quantita), not in length, for even though both of them are indivisibles, the line BC will be as much larger than the line BA as the latter is longer than the former. 11
e'
セ
e
I
r
V'
Oセ
jB
A
lA' I
I
•..
,
B
]
G'l
セZh
GbOセ
V/ I
E
A
D'
1
D
Figure 3.
As Torricelli himself showed, this new notion of indivisible not only solves paradoxes, but also is very helpful to determine tangents. Let AB'B (see Figure 3) be the general parabola
セZケ
o
e:T'
In order to determine the point E such that EB is the tangent at B, complete the rectangle EFBD and consider the lines FB and BD, which are equal «in quantity,) as just shown. To determine the ratio FB: GB, which is the same as ED; AD and so will give us the point E, Torricelli points out that the ratio between FB and GB is the same both «in quantity» and in length. Whence ED
BD «in quantity»
AD
GB «in quantity»
To determine this last ratio, Torricelli assumes that BD and GB are reached by infinite division. That makes this ratio equal to m/n. Knowing that the figure BB'D'D is ro the figure BB'G'G as are the exponents m, n, he takes the middle point 1 on D'D and poinrs out that the figures BHID and BH]G are in the same ratio. So he can conclude, if we make this division, or assume thar ir has been made, an inhnire number of times, rhere will remain, instead of hgures, two lines [BD] and [BG] rhar, nor in length, but in quamity will be as rhe exponem [m] is to rhe exponent [n].12
Figure 2.
10. ¡bid., 3201. Cf. Bortolotti, ,,1 progressi del merado infinitesimale". 11. ¡bid., 322.
G
F
Were all the inhnite lines parallel ro the base AC drawn, the segment [PQ] would mark as many points on the straight line AB as in the straight line BC; whence any point of the former is ro any point of the latrer as the whole line [AB] is ro the whole line [Bq. 10 Torricelli did not drop the word «indivisible», but the notion he used here is by no means the same one Cavalieri had in mind. Furthermore, in Torricelli's hands indivisibles became not only comparable among themselves, they became reachable by infinite division as well, thus allowing the mathematician ro translate ratios between figures into ratios between indivisibles. For instance, starting from the obvious equality of the trapezoids EBAA' and EBCC' (see Figure 2), he would point out that, by taking the middle point 1 ofEB, two trapezoids IBA, IBC, which were equal as well, were obtained. So,
Remaking Indivisibles: Pascal, Barrow. Wallis 27
12. ¡bid, 3223 .
28
From Indivisibles
to Infinitesimals
Antoni Malet
Since 1'orricelli's indivisibles have no precedem in classical mathematics, securing their soundness seemed a necessary condition for the method of indivisibles to beco me an standard mathematical too!' 1'orricelli himself seems ro have thought as much, although the most imeresting views on the marrer carne from Pascal and Barrow. Pascal
Pascal justified the use of «indivisibles» by explicidy transforming them imo infinitesimals, and then dismissing any suggestion that the larrer notion involved logical difficulties. He said, in the wellknown words ofhis 1658letter ro Carcavi, je ne feray aucune difficulre d'user de cerre expression; la somme des ordonnees, qui semble n'esrre pas Geomeuique a ceux qui n'emendem pas la doctrine des indivisibles, er qui s'imaginem que c'esr pecher comre la Geometrie que d'exprimer un plan par un nombre indefiny de lignes; ce qui ne vient que de leur manque de imelligence, puisqu'on n'emend auue chose par la, sinon la somme d'un nombre indefiny de recrangles fairs de chaque ordonnee avec chacune des perires ponions egales du diametre, dom la somme est cenainemenr un plan, qui ne differe de !'espace du demy cercle que d'une quamire moindre qu'aucune donnee. 13
Pascal was not in the least willing to conclude that a surface was the aggregation of all of its ordinates. He saw no problems, however, in understanding it as made up by infinitely many rectangles with infinitesimal bases. We shall come back ro his comemion that this infinite sum would yield the surface of the figure with undetectable error. Pascal's thoughts on indivisibles and infinitesimals are fully set forrh in the opuscule «De l'esprit géométrique», undated but probably written in 1657 or 1658. 14 Geometry's objects are, according to Pascal, motion, number, and space things that encompass the whole universe, according ro the Holy Writ, and which have a «reciprocal and necessary relationship». 1'0 them time must be added, because time and motion are umhinkable without each other. 15 Knowledge of what belongs in common to all these notions, therefore, «ouvre 13. Oeuvres, VIII, 3523. 14. "De l'esprit geometrique», in Oeuvres, IX, 24070, particularly 256ff. On Pascal's indivisibles, see H. Bosmans, "La notion des indivisibles chez Blaise Pascab, Archivio di Storia della Scienza, IV, 1923,36979; ].L. Gardies, Pascal entre Eudoxe et Cantor (Paris: Vrin, 1984); J. de Lorenzo, "Pascal y Jos indivisibles», Theoria, 1, 1985,87120. The interpreraríon here offered is at odds with Bosmans's and Gardies's. 15. ,,[Mollvement, nombre, espace], qui comprennent tout l'universe, selon ces paroles: Deus ficit omnia in pondere, in numero, et mensura (Sap. XI, 21), Ont une liaison reciproque er necessaire. Car on ne peut imaginer de mouvement sans quelque chose qui se meuve; et cene chose etant une, cette unite esl ['origine de tous Jes nombres; enEn le mouvement ne pouvant estre sans espace, on voit ces trois choses enfermees dans la premiere. Le temps mesme y est aussi compris: car le mouvemem et le temps sont relalifs l'un a l'aune; la promptitude et la lenteur, qui SOnt les differences des mouvemems, ayant un rapport necessaire avec le temps.» From B. Pascal, "De l'esprit geometrique", in Oeuvres, IX, 24070, p. 256.
Chapler 2
Remaking Indivisibles: Pascal, Barrow, Wallis 29
l'esprit aux plus grandes merveilles de la nature». 1'he most imponam thing they have in common is infinity: La principale (propriéré commune] comprend les deux infinirés qui se rencon-
uem dans coures: l'une de grandeur er l'aurre de peritesse. 16
Pascal considers the two infinities to be imimately related, «de telle sone que la connaisance de l'un mene necessairement ala connaisance de l'autre.»17 He also assumes that the actual infinity of space emails its subdivision in actual infinitesimal pans: De sone que l'augmemation infinie enferme necessairemem aussi la division infinie. Er dans I'espace le mesme rappon se voir emre ces deux infinis comraires: c'est a dire que, de ce qu'un espace peut esue infinimem prolongé, il s'ensuir qu'il peut esue infinimem diminué, ... 18
Pascal adduces here an example that involves an optical device. Imagine that through a telescope we observe a boat gerring away in a straight lineo 1'he spot on the glass where one specific poim on the boat is observed would go cominuously up (haussera toujours par un flux continueb as the vessel would go away. If the boat goes to infinity, the spot will go up and yet it will never reach the poim where a horizomal ray coming from the eye reaches the telescope. D' ou l'on voit la conséquence necessaire qui se tire de la infiniré de l'erendue du cours de vaisseau, a la division infinie er infinimem perite de ce perir espace resram au dessous de ce poim horizomal. 19
Elsewhere he used another optical analogy. He stressed that pans so tiny as ro be invisible (imperceptibles) can be enlarged by lenses umil they equal the firmamem which shows that a litde space may have as many fcans (in particular, we may add, an infinite number of pans) as a large one. o 1'hese considerations, however, were but bagatelles. 1'he basic argumem rested on the infinite divisibility of space. 1'his notion assumed by all geometers, says Pascal, emailed the existence of infinitesimals unless the end product of infmite division be truly indivisibles. 1'herefore Pascal endeavored ro demonstrate that indivisibles cannot be reached by division, nor can they be pans of a cominuum.
16. 17. 18. 19. 20.
¡bid. ¡bid., p. 261. ¡bid. ¡bid. ¡bid., p. 258. He concluded by poiming out rhat nOlhing assures us that we see objects in its «lme» (veritable) size, for lenses may have reestablished the (fue size «que la figure de notre oei] avait changee et raccourcie».
30
From Indivisibles to Infinitesimals
Anroni Malet
He rehearsed two of the most well-known argumems against a cominuum made up of indivisibles. Two indivisibles cannot make up extended space because this entails they must touch each other. If they touch everywhere, they cannot be two different indivisibles. If they rouch in part, they have parts, so they are not indivisibles. The second classical argument goes against the notion that space is made up of a finite number of indivisibles. If this were rhe case, then there cannot exist an square the surface of which doubles the surface of a given square. 21 He also stressed the heterogeneous (sic) characrer of indivisibles vis-a-vis the cominuum. Ir is obvious rhat any space, howsoever small ir be, can be halved, and the parts obrained halved again. How could ir be that at some poim two halfparts turned out ro be indivisibles? This is absurd, because then two indivisibles wirhout extension (etendue), taken rogether would make up extended space. Pascal highlighred the evidence and clarity of this conclusion: Il n'y a point de connaissance naturelle dans l'homme qui precede cellesla, et qui les surpasse en dane. 22 Why is it, then, that excellem minds deny the existence of > (Ph.D. Diss., Princeton U niversity, 1989), Appendix to chaprer l. 30. See Newron's Mathematical Papers, Ir, 22232. 31. In 169192, rhat is; cf. Newton's Mathematical Papers, VII, 48 nI.
Opposing Indivisibles: Huygens, Gregorie, Newton 61
A second difference between Newton's De Analysi and the Exercitatio is found in how the two works prove the basic lemma that we express x fax"'/ndx = Nセ :!m+n)/n, m,n being positive integers. [1] O m+n
To prove this result Newton assumed x (see Figure 4) to be equal to z = 2.. >.3/2.
= AB and a surface such as ADB
3
For a different x, say AC or x + o, the surface z becornes z + ov, v being sorne interrnediate value benveen the ordinates BD and CE. Therefore, ; (x + 0)3 = (z + ov)2.
Newton concludes, after developing parentheses and canceling, lf we now suppose [Be] ro be infinite1y small, rhar is, o ro be zero, v and y [rhe ordinare DB, rhar is] will be equal and rerms mu1tiplied by o will vanish and rhere will consequendy remain ... x l / 2 = y. Conversely rherefore if x l / 2 = y, rhen will _2_ x)/2 = z.
3
Finally, Newton explained why the argurnent works for any rational positive exponent of x. 32 In the Exercitatio Geometrica the proof of the result [1] is referred, for m/n positive, to Proposition 54 of the Geometriae pars universalis, and for m/n negative to Proposition 59 of the sarne. We shalllook in detail at the first of these theorems. Let rhe curve ICA (see Figure 5) be defined by AB _( AK
EC
)
)P/q ,
AE
AK, AB being the sides of a given rectangle, and DE being any parallel ro AB. Proposition 54 of the Geometriae pars universalis states that the whole paralle!ogr am ABIK is to the figure ACIK as p + q to q. To prove this result, Gregorie mtroduces an auxiliary curve AHG defined by taking, for every C on the curve セcaL CH (parallel to KA) equal to the subtangent EF at C. Now, says Gregorie, 1t has been proved in Proposition 11 that Figure ACIGHA = Figure ACIK. 32. Newron's Mathematical Papers, !l, 2424 (Whiteside's translation).
62
Amoni Malet
From Indivisibles ro Infinitesimals
Q
J
セ
Opposing Indivisibles: Huygens, Gregorie, Newton
63
figure ACIGHA as pro q, whence rhe whole recrangle Al, which is equal ro ACIGHA plus ABICA, is ro ACIGHA, or AClKA, as p + q ro q.33 Ler us look now ar Proposition 11, which plays a crucial role in rhe foregoing proof and provides an interesring relarionship between quadrarures and rangenrs generally.34 Proposirion 11 esrablishes, for any curve wharsoever ACI (see Figure 5), rhar figure ACIK = figure ACIGHA, where rhe curve AHG has been defllled by drawing from every point C a segment CH parallel ro AK and equal ro rhe subrangent EF ar C. Gregorie's exhausrion proof is grounded in rhe following inequaliries:
s
rrapezoid ICEK > mixtilinear figure ICHH' recrilinear figure IMCEK mixrilinear figure ICHH' trapezoid CAE > mixrilinear figure CAL,
F and also
B
A
E
K Figure 5.
On rhe orher hand, rhe subrangent EF of rhe curve ICA is always found by raking AE : FE :: p : q. (Gregorie does nor explicirly deduce rhis result, bur says ir is a consequence of rhe general merhod ro find rangents he has ser forth in Proposirion 7.) Therefore, by Cavalieri's principie, rhe figure ABICA is ro rhe
recrilinear figure IMCEK B. 18. As shown in Figure 20, C and L are assumed ro be on rhe straight lines QP and ZP, respectively.
90
From Indivisibles to Infinitesimals
Anton; Malet
A: 13 > PC : PQ.19 Then, from C, let the tangent CF to the curve be drawn, and [from Q] let a straight line parallel to ir and meeting the curve at E be drawn. Through E let the straight line PEO, which meets CF produced at O, be drawn. Then, as ir has just been done, let the tangent OIL, from O, and its parallel EHM be drawn. This is ro be repeated until finally the whole curve is surrounded by the tangents. 1 say, that the ratio of the circumscribed srraight lines CO + OL to the inscribed ones QE + EM is less than the ratio of A to B. By construction A: 13 > PC: PQ. Bur PC : PQ:: CO : QE :: PO : PE :: OL : EM, whence PC : PQ :: CO + OL : QE + EM. That is, [theyare] in a ratio less than rhat of Ato B. Q.E.D. PROPOSITION 8. PROBLEM J say, the same things being assumed, that the circumscribed fines are greater than the curve and the inscribed ones less than it. FO + DI > curve FI (see Figure 10);20 likewise F4 + Q4 > curve QF. But C4 > Q4, because the angle CQ4 is greater than an acure one and therefore the [angle] QC4 is to that extent less than an acute one. Ir is therefore C4 + 4F > Q4 + 4F, and C4 + 4F, or CF > curve QF. Ir is proved similarly that IL > curve IZ. Whence FO + DI + CF + IL > curve FI + curve FQ + curve IZ, that is, CO + OL > curve QZ. Q. E. D. Now, curve QE > straight line QE. 2 1 Since the angle LZE [read IZ3] is greater than an acure one, of necessity [angfe] ZL3 = angle PMH is acute. Whence it follows that the angle HML is obtuse and EZ > EM. But the curve EZ > the srraight line EZ, hence the curve EZ > the straight line EM, and curves QE + EZ> straight lines QE + EM. That is, the curve QEZ is greater than the straight lines QE + EM. Q.E.D. PROPOSITION 9 J say, the same things being assumed, that the circumscribed fines can be greater than the inscribed ones by less than any given lineo
James Gregorie', "Sorne General Proposirions of Geomerry" 91
Chapter 4
Let curve QEZ = a (see Figure 10) and the given line = Y. Let PC : PQ QE + EM. Ir is, therefore, CO + OL QE EM : QE + EM < Y : QE + EM. That is, Y > CO + OL QE EM. Q. E. D. PROPOSITION 10. THEOREM
Similar curves are as corresponding straight lines. Let QEZ, AVK be similar curves (see Figure 10); 1 say, that they are ro one another as corresponding srraight lines. Inside the curve QEZ let a point P be raken such thar each of rhe angles CQ4, LZ3 is greater than an acute one, and 23 let this point P be the common corresponding point. 1 say, that curve QEZ : curve AVK :: srraight line PQ: [straight line] PA. If it is denied, let PQ: PA :: a: AVK, and let the difference between a and QEZ be equal to Y. According to the preceding result, let us take CO + OL _ QE _EM a. Similady, [since] QP : AP:: QE + EM: AV + VY, therefore QE + EM : AV + VY:: a : [curve] AVK and alternando QE + EM : a :: AV + VY: AVK, but AV + VY ,
Chapter 4
97
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drawn from point G through [a point in] the perimeter of the figure NLM a1so goes through [a point in] the perimeter of the figure KIH. Therefore (according ro the e1eyenth proposition aboye) the figures KIH, NLM are in the duplicate ratio of that between the straight lines GL and GH. Q.E.D.
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Similar sofids are in the tripficate ratio o/that between corresponding straight fines.
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PROPOSITION 14. THEOREM Let two simifar sofids be cut by paraffefplanes going through the extreme points o/ corresponding straight fines. 1 say, that the pfane figures cut, or the common intersections, are in the dupficate ratio o/that between corresponding straight fines.
Let GABC, GEDF be two similar solids (see Figure 12) cut by parallel planes KIH, NLM that go through L and H, the end points of corresponding straight lines GL, LH. I say, that the figures KIH, NLM are in the duplicate ratio of GL toHG. Let any point in the perimeter of the figure NLM, say M, be joined by the straight line GM to the common corresponding point G, and let GM be produced until it meets the other plane at 1. It is manifest, because the planes are parallel, that GL : GH :: GM : G1. Because of this (and since the solids are similar), the point I is also on the surface of the solid GABC; as a consequence, I is also in the perimeter KIH. For the same reason, eyery straight line
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Let A and B be similar solids (see Figure 13), O their common corresponding point, and DE, DF corresponding straight lines. I say, that A is to B in the triplicate ratio ofDE to DE If the solids A and B are not in the triplicate ratio ofDE to DF, let A be to X in the said triplicate ratio of DE to DF, and let Z be the difEerence between B and X. Let right cylinders with the same height be inscribed in and circumscribed about the solid B, and let the difference between the inscribed cylinders and the circumscribed ones be less than Z. Let MC be one oE the inscribed cylinders and MK a circumscribed one. From the perimeters of the bases oE the cylinders let straight lines CD, LO, etc, which cut the surface oE solid A at N, H, etc, be drawn. Let planes parallel to the bases oE the cylinders EN, IH, etc, be drawn to intersect the solid A; then having for bases these intersections, \\ [7] the cylinders IN, IG, etc, inscribed in and circumscribed about the solid A are to be completed. It is maniEest that the plane EN is to the plane FC in the duplicate ratio oE that between the corresponding straight lines DE and DE Similarly, the figure IH is to the figure ML in the duplicate ratio of OH to DL, or DE to DE Let us take aplane that runs through [the sofids A and E], cuts perpendi-
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figures ABT, ABe wirh heighr AB equal rhe solids ABT, ABC, respecrively. Now rhe resulr foll ows from Proposirion 3. 28. Thar is, rhe inrersecrions on rhe same plane.
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From Indivisibles
tO
Infinitesimals
Antoni Malet
, p. 354355. T¡'e Leibniz-Clarke eorrespondenee' (Manchesrer: Manchesrer Universiry 3. H.G. A1exander Press, 1956), p. 48.
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From Indivisibles ro Infinitesimals
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Anroni Malet
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Apparently, these were thoughts widely held. In 1669 an obscure correspondent ofHuygens, the Baron de Nulandt, conveyed to him similar feelings: I'infiní et le rien a peine peuuene ils estre cODl;:eus l'un sans l'autre, car ce qui est infini a l' esgard de quelque chose, cette mesme chose est rien a l' esgard de \' autre, de sone que rien a la mesme propon ion a quelque chose finie, que cette mes me chose a a l'infini; ... J' aij remarque de mesme que ces noms sone aequiuoques et relatifs et que la mesme chose peut estre infinie a l'esgard d'une, et rien a l' esgard de quelque autre chose, comme une ligne donne est infinie a I'esgard d'un point et touteffois est rien a l'esgard d' une surface; ... 5
We have seen in Chapter 2 that the old problem of the composition of continua in terms of indivisibles became irrelevant as infinitesimals were substituted for indivisibles. Almost simultaneously, as we shall see in the present Chapter, arguments traditionally regarded as formidable objections to the handling of infinites large and small seem to have lost their punch sometime during the second half of the seventeenth century. Timehonored cautions and theological connotations traditionally linked to discussing infinites faded quite suddenly as well. This, I would suggest, was at least partially a consequence of the majar role infinites and infinitesimals carne ro play in seventeenthcentury natural philosophy. The mechanical philosophy relies on the tacit assumption that each partide's essential (mechanical) properties its size, mass, inertia, and so onare independent of the existence of other partides. 6 The mere addition of partides's mechanical magnitudes accounts for the properties of macroscopic bodies. As there is no longer a whole body that bestows properties on its parts, so it is possible to consider the infinitesimal magnitudes of the parts independently of the magnitude of the whole. Even while the handling of individual partides ar the actual measuring of any of their magnitudes was obviously out of question and it was recognized explicidy to be so, the seventeenthcentury natural philosopher had no qualms assuming observable mechanical effects ro be equal ro the sum of very small magnitudes. In practice and mathematical computation, these very small magnitudes were handled as indivisibles or infinitesimals. A few years ago A. G. Molland argued that one feature of the Scientiflc Revolution was what he called «the atomization of motion». He convincingly 4. Pascal, "De ¡'esprit géométrique et de l'are de persuaden" p. 354. 5. F.W. de Nulandt to Huygens, 23 May 1669, in Huygens, Oeuvres, VI, p. 435. 6. G. Freudenthal has convincingly shown the essential rale this assumption plays in Newton's natural philosophy. See his Atom and Individual in the Age ofNeu'ton. On the Genesis ofthe Mechanistic World View, P. McLaughlin transo (Dordrecht: D. ReideI. 1986).
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Chaptet 6
Infinites and Infinitesimals in Seventeenth-Century Natural Philosophy
139
argued that analysis in terms of «moments» or «atoms» of time was typical of the seventeenthcentury, and so it was the view that motions and speeds were «composed of a uniquely determined set of ultimate elements»,7 Galileo provides the obvious example, but many others may be added. As the fol1owing exampie from Isaac Beeckman's diary indicates, the atomization of motion could be used to obtain many sorts of results, sorne ofwhich are not deemed correet roday. In 1618 Beeckman analyzed free fall by looking at it as constituted by successive motions through the indivisible moments making up the time of fall. He assumed that in every moment the body would describe a "parallelepiped» (sic) that would increase in length as the body's falling speed increases. When the quantity of air in the parallelepiped equals the body's weight, Beeckman argues, the force pulling the body down will be balanced by a contrary pull of the same magnitude coming from the air apparently applying in sorne innovative way Archimedes's principie ro the airo This demonstrates, he concludes, the existen8 ce of an «equality poino) in which the body gets a uniform falling speed. With the advent of the calculus in the late seventeenth century, M.S. Mahoney has convincingly identified a «process of bootstrapping by which mathematics and mechanics assisted and provoked one another ro deeper sophistication».9 The process, we may add, had strong roots in the way indivisibles and infinitesimals were used ro gain quantitative results in many fields, not only in motion, ever since the first decades of the century. In a famous passage, Galileo argued from the observable fact that ants carry wheat grains that a multitude of ants can pul1 ashore a big boat loaded with wheat. But this was merely introducrory stuff to his treatment of the paradox involved in the socal1ed Arisrotle's wheel, which in tum al10wed him to argue that an infinite number of minimal forces provides cohesion ro solid bodies. ID He also argued that particles in fluids must be something like mathematical indivisibles, and so must be the minimum particles of fire and of the sun's rays.11
7. A.G. Molland, «The atomisation of motion», Studies in History and Philosophy ofScience, 13 (1982): 3154; quotation comes fram p. 43. 8: e. de Waard ed., Journal ten upar Isaac Beeckman de 1604 a1634, 4 vol. (La Haye: Martinus Nijhoff, 1939), 1, [UVRセT quoted in R. Dugas, La mécanique au XVIIi: sii:cle (des antécédents scolastiques a la pensée classique) (Neuchátel: Éditions du Griffon, 1954), 527528. For a further example of «atomio> analysis of motion, this time fram Descartes, see ibid., p. 528529. 9. M.S. Mahoney, «Infinitesimals and transcendent relations: The mathematics of moríon in the late seventeenth century», in D.e. Lindberg, R.S. Westman eds., Reappraisals ofthe Scientific Revolution (Cambridge: Cambridge University Press, 1990), 461491, p. 484. 10. Discorsi e dimostrazioni matematiche, in torno adue nuoue seienze, in Opere, 20 vol. (in 21), A. Favaro ed. (Repr. Firenze: Barbera, 1968; 1st ed., 18901909), VIII, 39346, p. 67ff. 11. Discorsi, in Opere, VIII, p. 85; see also Opere, IV, p. 106. The role this view of fluids play in Galileo's physical arguments has been pointed out in M. Biagioli, «Anthropology of Incommensurability», Studies in History and Philosophy ofScíence, 21 (1990), 183209. For the indivisibles of fire and light, see Opere, VIII, p. 86; VI, 347352; and his comments to Baliani's lener of 8 August 1619, in Opere, XII, p. 475. On Galileo's corpuscularism, see W.R. Shea, Galileos intellectual revolution (London: Macmillan, 1972), p. 2731,101105.
140 From Indivisibles ro Infinitesimals
Anroni Malet
Tuming to others authors, Huygens decomposed surfaces and bodies in infinitesimal e1ements in order to calculate centers of gravity and centers of oscilIation. 12 Another, less familiar example of the application of indivisibles is James Gregorie's use in optics of a typical Cavalierian technique. In his 1663 Optica promota, containing the first attempt to quantifY optical devices, Gregorie used indivisibles ro measure «the strength to bum or illuminate» of radiation cones coming from a radiating body. First he found a proportion between the strength to bum or illuminate of two radiation cones coming from a radiating point A and the squares of a trigonometric function of their angles at the vertex. Next, he proceeded ro «divide the radiating body in its radiant points». Finally, by taking «all the antecedents» and «all to consequents» he extended the proportion to radiation cones coming from the whole radiating body.13 This is the most wellknown use natural philosophers made of infinites, infinitesimals and indivisibles through the seventeenth century but it was by no means the only one, nor perhaps the most decisive one. As we shall see presently, infinites and infinitesimals were also used to define notions and characterize hypothetical substances. Huygens, for instance, characterized the force of percussion as being infinite (infinie) visavis static force. 14 Key notions of the mechanical philosophy such as Huygens's atoms and gravitational ether, Hooke's menstruum, or Newton's empty space were characterized byendowing them, or the particles constituting them, with sorne physical property (fluidity, ve1ocity, hardness, emptiness, .. ) in an infinite degree. In Book nI of the Principia mathematica Newton's (empty) space is pro- ved as a corollary ofProposition VI (the weights ofbodies, at equal distances from the center of the planet, are proportional to their quantities of maner). Corollary In asserts that «all spaces are not equally fuI!». Therefore, «if the quantity of maner in a given space can, by any rarefaction, be diminished, what should hinder a diminution to infinio/» (italics added). Next, assuming that bodies are made up, in the last analysis, of solid particles of the same den- sity that «cannot be rarefied without pores», Newton concludes in Corollary IV that «a void, space or vacuum must be granted». 15 So (infinite) emptiness is conceived by opposition ro absolute fullness. 12. C. Huygens, Oeuvres completes, XVI, 384555, passim. 13. Proposition 33 of Gregorie's text establishes that "Vites Conorum tadiosotum, in illus- ttando, ve! comburendo (radiis nimirum in spatia aequalia absque debilitatione congrega- 6s) sunr in duplicata ra60ne Chordarum, suorum semiangulorum radiosorum.» Corollary 1 exrends rhe result, first ptoved for points, to a "corpus radiosum" by the rypica1 expedient oF